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Carbon nanotubes in passive rf applications

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									                                                                                          20

    Carbon Nanotubes in Passive RF Applications
                                         Ahmed M. Attiya and Majeed A. Alkanhal
                                         King Saud University, Electrical Engineering Dept
                                                                             Saudia Arabia


1. Introduction
Carbon nanotubes are characterized by unique electrical properties which make them good
candidates for different applications in electronics and electrical engineering. In this chapter
we focus mainly on electrical properties of single wall conducting carbon nanotubes in high
frequency, electromagnetic waves interaction with carbon nanotubes and the possible
passive RF applications. The term “high frequency” here refers to the frequency band from
gigahertz to terahertz. This chapter starts from microscopic view by discussing
electrodynamics of carbon nanotubes to show the mechanism of time varying
electromagnetic field interaction with carbon nanotubes (Slepyan et al., 1999; Slepyan et al.,
2008; Mikki & Kishk 2008). Based on these electrodynamics properties, an equivalent
dynamic surface conductivity is developed to represent a macroscopic view for the
interaction of high frequency electromagnetic fields with carbon nanotubes (Hanson, 2005).
This equivalent surface conductivity of carbon nanotube is characterized by complex value
with negative imaginary part. This negative imaginary part represents an inductive effect in
carbon nanotubes. This inductive effect is due to chiral property of the electric current flow
along the carbon nanotube (Slepyan et al., 1998; Miyamoto et al. 1999). This inductivity has a
significant effect on reducing the wave velocity of electromagnetic wave propagation along
carbon nanotube. This wave velocity reduction corresponds to decreasing the wavelength.
This property is quite important in passive RF applications like passive circuits and
antennas, since the dimensions of these applications depend mainly on the wave length
(Slepyan et al., 1999; Slepyan et al., 2008; Attiya, 2009).
Based on the macroscopic surface conductivity of carbon nanotube, the problems of
electromagnetic fields interaction with carbon nanotubes can be presented in similar ways to
conventional problems related to cylindrical structures with finite surface conductivity. In
this way the problem of carbon nanotube antennas can be presented as an electric field
integral equation problem which can be treated numerically by method of moments
(Hanson, 2005; Hao & Hnason, 2006). Similarly, the problem of surface wave propagation
along carbon nanotubes can be presented as a boundary value problem where the difference
between the tangential magnetic fields on the two sides of the wall of the carbon nanotube
would equal the induced current on the wall of the carbon nanotube. This induced current
depends on the tangential electric field along the carbon nanotube and the surface
conductivity. This boundary value problem is solved to obtain the field distribution and the
complex propagation constants of the surface wave modes propagating along carbon




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472                                             Carbon Nanotubes Applications on Electron Devices

nanotube (Slepyan et al., 1999; Shuba et al. 2007; Slepyan et al., 2008; Attiya, 2009). This
surface wave propagation is characterized by highly attenuation coefficient at microwave
frequency band. This property makes carbon nanotubes are not suitable for antenna design.
However, this high attenuation property is more suitable for other applications which are
based on absorbing or attenuating electromagnetic waves like transparent electromagnetic
shielding (Xu et al., 2007) and microwave heating in biomedical applications (Mashal et al.,
2010). On the other hand, at higher frequency bands in the range above 100 GHz, this
attenuation coefficient is decreased and carbon nanotubes can be a good candidate to design
low loss antenna structures of much smaller size compared with operating free space wave
length (Huang et al. 2008 & Attiya, 2009).
Another common approach for simulating electromagnetic wave propagation along carbon
nanotube is based on electron fluid model (Burke, 2002; Chiariello et al., 2006a; Miano &
Villone 2006). This model is more suitable for simulating transmission line sections of
carbon nanotubes. In this case the inductive effect of current flow along the carbon nanotube
transmission line is modeled as an additional kinetic inductance in the equivalent circuit
model of this transmission line (Burke, 2002; Chiariello et al., 2006b; Miano & Villone 2006;
Maffuci et al. 2008; Maffuci et al. 2009). This kinetic inductance has much greater value than
the conventional magnetic inductance of conventional transmission lines. This increase in
the total inductance introduces two main effects; decreasing the wave velocity along the line
and increasing the characteristic impedance of the line. To improve the properties of carbon
nanotube transmission lines, bundles of carbon nanotube are used instead of a single carbon
nanotube (Plombon et al., 2007; Rutherglen et al. 2008). Extensive studies are presented in
literature about the possibility of using carbon nanotube bundles as interconnects in high
speed integrated circuits (Massoud & Nieuwoudt, 2006; Naeemi & Meindl, 2009).
Recently, another new approach is discussed for solving the interaction between Maxwell’s
equation and Schrödinger equation numerically by using finite difference method to obtain
electromagnetic field interaction with nanodevices like those which are based on carbon
nanotubes (Pierantoni et al.; 2008, Pierantoni et al.; 2009 & Ahmed et al.; 2010). This method is
based on space-time discretization. The electromagnetic source is modeled by means of time
dependent vector and scalar potentials which are added to the quantum potential profile of
the carbon nanotube. Then Schrödinger equation is solved by using a finite difference
scheme to obtain the wave equation of electron flow along the carbon nanotube.
The aim of the present chapter is to introduce to the reader an updated view for the
problems of electromagnetic field interaction with carbon nanotube with emphasis on the
possible passive RF applications. Section 2 presents the electrodynamics of carbon nanotube
and the concept of equivalent macroscopic surface conductivity. Section 3 presents the
electron fluid model of carbon nanotube and how it can be used to obtain equivalent circuit
parameters of carbon nanotube transmission lines. Section 4 presents finite difference time
domain method as numerical technique for studying electromagnetic interaction with
carbon nanotubes. Section 5 presents detailed analysis of surface wave propagation along
carbon nanotubes. Section 6 introduces the electric field integral equation formulation of
carbon nanotube antenna and presents sample results for this problem. Section 7 discusses
the possibility of using carbon nanotubes in some passive RF circuits. Finally, Section 8
introduces some possible applications of carbon nanotubes based on their absorbing
properties in microwave frequencies.




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Carbon Nanotubes in Passive RF Applications                                                 473

2. Dynamic conductivity of carbon nanotubes
Dynamic conductivity of a carbon nanotube represents a macroscopic quantity relating the
disturbance of electron flow along the nanotube due to an incremental temporal variation in
the applied electric field along it. For conventional carbon nanotube structures, the length of
the nanotube is much greater than its circumference. Thus, for most practical cases, it is
assumed that the equivalent current along the surface of the nanotube is transversely
symmetric and parallel to the axis of the nanotube. In the following analysis the geometry of
the nanotube is assumed to be presented in cylindrical coordinate system, where the axis of
the nanotube lies along the z-axis. Thus, the proposed dynamic conductivity in this case is
the relation between the surface current density J z and the applied electric field Ez .
The applied field is presented as a time harmonic propagating wave along the axis of the



                                                                       
nanotube as follows:

                                      Ez ( z , t )  Re Ez e jt  z
                                                         0
                                                                                             (1)

where Ez is the amplitude of the incident field,  is the angular frequency and  is the
         0

complex propagation constant along the nanotube. This complex propagation constant is
discussed in detail in Section 5. However, in the present case, the dependence of the electric



                                
field on z can be assumed to be constant by taking the limit where an incremental length of
the nanotube is considered. Thus, for a very small part of the nanotube, the incident field is
assumed to be Ez (t )  Re Ez e jt .
                             0

This applied electric field introduces a disturbance in the electron distribution function
along the nanotube. At equilibrium, the electron distribution function is given by:

                                F(p)  1 / 1  exp  (p) / kBT  
                                                                                           (2)

where p  pza z  p a is the electron’s two-dimensional quasi-momentum, kB is Boltzman
constant, T is the absolute temperature and (p) is the electron energy with respect to the
Fermi level in the lattice of the carbon nanotube. By applying an axial time harmonic electric
field on the nanotube, this distribution function along an incremental length is modified as



                                                                           
follows:

                                     f ( p, t )  F(p)  Re  f e jt                        (3)

This dynamic distribution function is governed by Boltzman kinetic equation (Dressel &
Grüner, 2003). By taking into consideration that the problem is transversely symmetric, the
applied field is only along the axis of the nanotube one can obtain Boltzman kinetic equation
for the carbon nanotube as follows:

                                 f       f       f 1
                                     eEz      vz    F  f 
                                 t       pz      z 
                                                                                             (4)

where  is the electron relaxation time in the lattice of the carbon nanotube, v z  (p) / pz
is axial electron velocity and (p) is the electron energy function. This relaxation time is
nearly 3 ps in carbon nanotube (Hanson 2006). By applying the time harmonic electric field




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474                                                        Carbon Nanotubes Applications on Electron Devices

and the dynamic distribution function of Equation (3) into Equation (4), one can obtain the
incremental disturbance in the distribution function as

                                                        jeEz F
                                                f 
                                                           0

                                                         j pz
                                                                                                        (5)

where v  1 / is the relaxation frequency and e is the electron charge. The amplitude of
the time harmonic current density along the surface of the nanotube can be obtained by
using this disturbance function in the electron distribution function as follows:

                                        Jz           vz f dpzdp
                                         0      2e
                                                                                                        (6)
                                                h 2 1st BZ

where h is Planck’s constant. The range of integration in Equation (6) refers to the first
Brillouion zone of the carbon nanotube lattice. By inserting Equation (5) into Equation (6),
one can obtain a linear relation between the amplitude of the incident time harmonic electric
field and the resulting surface current density as follows:

                                                  J z   zzEz
                                                    0        0
                                                                                                        (7)
where the equivalent axial conductivity  zz is given by


                                                                                   F  (p) 
                               pz pz                                        pz  dpzdp
                                    F(p) (p)
          zz                                  dpz dp  2                                  
                                                                                          2
                  2 e2                                   2 e2
                  h   j                               h   j
                       j                                      j
                    2
                                                                                                        (8)
                             1st BZ                                         1st BZ

It should be noted that the azimuth momentum p has discrete values in nanotube since
electron energy in this case is a periodic function of  . Thus, the double integration in
Equation (8) is converted into a finite series of single finite integration.
The key difference between the conductivities of the different types of the carbon
nanotube lies in the corresponding electron energy function E(p). This electron energy
function depends mainly on the chiral vector of the carbon nanotube. Each chiral vector is
a combination of integer multiplications factors, m and n, of the two basis lattice vector for

the carbon nanotube lattice are m  0 and n = 0 and, The electron energy function is given
a graphite sheet. For a zigzag carbon nanotube, where the indices of the chiral vector of

by:

                                                 3bpz           3bp             3bp 
                   zigzag (p)   0 1  4 cos                 
                                                            cos         4 cos 2       
                                                 h /                              h / 
                                                                 h /                  
                                                                                                      (9-a)


where b  1.42 A is the interatomic distance in a graphite sheet,  0  2.7  3.0 eV is the
                  


characteristic energy of the graphene lattice and the azimuth momentum in this case is



                                                      
given by:

                                p  hs /       3mb ,         s  1, 2, 3,....., m                    (9-b)




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Carbon Nanotubes in Passive RF Applications                                                       475

For the case of an armchair carbon nanotube where the indices of the chiral vector are equal,
m=n, the electron energy function is given by:

                                                3bp       3bpz            3bpz 
                armchair (p)   0 1  4 cos        cos          4 cos2 
                                                             h / 
                                                                              h / 
                                                                                     
                                                h /                            
                                                                                                (10-a)


where the azimuth momentum in this case is given by:

                                  p  hs /  3mb  ,         s  1, 2, 3,....., m              (10-b)

By inserting Equations (9) and (10) into Equation (8) and evaluating the required integration
one can obtain the axial conductivity for both zigzag and armchair carbon nanotubes. For
the cases of small values of m (where m<60), these integrals can be evaluated approximately
in closed forms. Zigzag carbon nanotubes have conducting properties for values of m which
are integer multiple of three. In this case, the dynamic conductivity of zigzag carbon
nanotube is given by:

                                                8 3 e 2 0
                          zz _ zigzag   j                               m  3N , n  0
                                               mh 2   j 
                                                              ,                                 (11-a)

On the other hand, armchair carbon nanotubes are always conductor for all values of m. The
dynamic conductivity of armchair carbon nanotube is given by:

                                                          8 e 2 0
                                 zz _ armchair   j                          mn
                                                        mh 2   j 
                                                                       ,                        (11-b)

For a chiral carbon nanotube where m  n and n  0, the carbon nanotube is conducting if
2m + n = 3N where N is an integer value. In this case the dynamic conductivity of the carbon
nanotube is given by:

                                                  8 3 e 2 0
                    zz _ chiral   j                                           2 m  n  3N
                                         h 2 m 2  mn  n2   j 
                                                                           ,                    (11-c)


Equation (11) represents the dynamic conductivity for the different types of conducting
carbon nanotubes. It should be noted that this conductivity corresponds to a surface
conductivity. Thus the unit here is Siemens. It can be noted that these dynamic
conductivities are complex values of negative imaginary part while for conventional
conductor the conductivity is usually a real part. This negative imaginary part in the
conductivity of carbon nanotube corresponds to an additional inductive effect in the
mechanism of the electron current flow along it. This inductive effect introduces slow wave
propagation along the carbon nanotube as it is discussed in the following section. This slow
wave property corresponds to a decrease in the wavelength along the carbon nanotube.
Since the dimensions of RF circuits and antennas depend on the electrical length, this
reduction in wave velocity along the carbon nanotube is expected to be quite useful for size
reductions of RF circuits and antennas.
For the sake of comparison (Hanson 2005) introduced the equivalent surface conductivity of
a hollow copper nanotube as:




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476                                                  Carbon Nanotubes Applications on Electron Devices


                                  2 d _ cu ( )   j
                                                         me   jvcu 
                                                                 2d
                                                           e 2 N e _ cu
                                                                                                 (12)


          N e _ cu  1.9271  10 19 electrons/m 2
             2d

me  9.1  10 31 kg is the mass of the electron and vcu  41.667 THz is the electron
where                                             is the surface electron density and

relaxation frequency of copper. Figure 1 shows a comparison between the dynamic
conductivity of armchair carbon nanotubes for different values of m. It can be noted that the
conductivity of the carbon nanotube decrease by increasing m. The imaginary part of the
conductivity is zero at dc and it has a beak value around 50 GHz. The real part of the
conductivity is decreasing by increasing the operating frequency.




are Re( ) ; dashed lines are Im( ) . (Hanson 2005).
Fig. 1. Dynamic conductivity of armchair carbon nanotube for various m values. Solid lines




Fig. 2. Comparison between the conductivity of an armchair carbon nanotube of m = 40 and
the conductivity of an infinitely thin copper tube of the same radius (2.712 nm) (Hanson
2005)
Figure 2 shows a comparison between the conductivity of an armchair carbon nanotube of m
= 40 with the surface conductivity of a hollow copper tube of the same radius. It can be
noted that the imaginary part of the copper conductivity is negligible from dc up to 1 THz.
Below 100 GHz, the real part of the conductivity of carbon nanotube is greater the
conductivity of copper. However, at higher frequencies both the real and imaginary parts of
the conductivity saturate at much smaller values.




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Carbon Nanotubes in Passive RF Applications                                                   477

3. Electron fluid model of carbon nanotube transmission line
In this section electron fluid model is presented as an alternative representation to describe
the linear response of a single wall metallic carbon nanotube to an applied electromagnetic
field. This method is based on presenting the problem in the classical form of moving point
charges in electric field with using appropriate effective mass for the moving electrons to
include the effect of the carbon nanotube lattice. The nanotube is modeled as a continuous

the z-axis as shown in Figure 3. In thermodynamic equilibrium the -electrons are
infinitesimally thin cylinder shell S of radius rc and length l. The cylinder is oriented along


distribution of the -electrons.
distributed uniformly where the applied electromagnetic field perturbs this equilibrium




Fig. 3. Carbon nanotube geometry

The collective motion of the perturbed  -electrons is modeled by considering them as a

the position vector of an arbitrary point on the surface S; n  n0   n( rs ; t ) is the surface
charged fluid. Assuming that v z ( rs , t ) is the mean velocity of the electron fluid, where rs is

number density of the electron fluid, where n0 is the equilibrium value; p  p0   p( rs ; t ) is
the “two-dimensional” pressure of the electron fluid, where p0 is the equilibrium value. The

by the relation  p  meff c s  n where cs is the thermodynamic speed of sound of the electron
incremental pressure perturbation is related to the incremental electron density perturbation
                             2

fluid if it is neutral and meff is the mean effective mass of the  -electrons. This
thermodynamic speed equals nearly electron Fermi velocity cs  vF  3 0b / h  8  10 5 m / s
The motion of  -electron fluid follows the law of momentum conservation which can be
presented in the present case as

                                        v z                   p
                                              vn0 meff v z        n0 eEz
                                         t                    z
                              n0 meff                                                         (13)


I z  2 rc env z and the surface charge density q  2 rc en as follows
This momentum conservation equation can be presented in terms of the longitudinal current



                                  I z           2 q   2 rc n0 e 2
                                        vI z  cs    
                                  t               z
                                                                     Ez                       (14)
                                                          meff

The parameter n0 / meff takes into account the influence of the atomic crystal field. This

semiclassical model based on Boltzmann equation as n0 / meff  4 vF /  hrc .
parameter is obtained for the case of a conducting armchair carbon nanotube by using

This representation corresponds to an equivalent distributed series RL per unit length with
shunt quantum capacitance per unite length as shown in Figure 4, where




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478                                                 Carbon Nanotubes Applications on Electron Devices

                                              I z           1 q
                                    Ez  Lk         RI z 
                                              t            C q z
                                                                                                (15)



Lk  h / 8e 2 vF , the quantum capacitance per unit length C q  8 e 2 / hvF and the ohmic
The elements of this equivalent circuit are the kinetic inductance per unit length

resistance per unit length R  vh / 8 e 2 vF respectively. Typical values of kinetic inductance
and quantum capacitance are LK  4nH/ m and CQ  400 aF/ m respectively.




Fig. 4. Circuit model for electron flow along carbon nanotube
It is interesting to note that the series elements of this equivalent distributed circuit of
carbon nanotube can be directly obtained by using the equivalent surface conductivity of
armchair carbon nanotube which is discussed in the previous section as follows:

                                     Zs               R  jLk
                                            2 rc zz
                                                1
                                                                                                (16)

However, the parallel quantum capacitance element cannot be obtained directly from this
surface impedance since we neglected the longitudinal derivative of the electron
distribution function in the derivation of the equivalent surface conductance. It would be
shown in the following discussion that the wave propagation on carbon nanotube is mainly
dominated by the kinetic inductance and the loss resistance. Thus, the approximation used
in deriving surface conductance does not have a significant effect on studying
electromagnetic wave propagation along carbon nanotube.
For the case of a carbon nanotube transmission line above a PEC ground plane as shown in
Figure 5, the equivalent distributed circuit would be a combination of the equivalent circuit
for electron current flow along the carbon nanotube and the conventional distributed
transmission line circuit which is based on electrostatic capacitance and magnetostatic
inductance of the transmission line structure. In the present case the distributed elements of
the equivalent circuit of the transmission line per unit length are

                                    LM  ( 0 / 2 )ln  2 d / rc                            (17-a)

                                    C E  2 0 r / ln  2 d / rc                           (17-b)


and  r is the relative permittivity of the supporting substrate. A typical value of substrate
where d is the separation distance between the carbon nanotube line and the ground plane

thickness is between 100 A and 1  m (Burke 2002) and Silicon dioxide substrate has a
                              o


dielectric constant  r  4 . For a Silicon dioxide substrate of 0.2  m thickness, typical values




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Carbon Nanotubes in Passive RF Applications                                                                           479


C ES  50 aF/ m , and LM  1pH/ m .
of single wall carbon nanotube transmission line equivalent circuit parameters would be




Fig. 5. Carbon nanotube transmission line above a PEC ground plane
In this case the effective inductance and resistance of the equivalent distributed circuit of the
carbon nanotube transmission line above a ground plane are given by

                                                                      
                                                   Leff   LM  Lk  / 1  C E / CQ                               (18-a)


                                                                  
                                                       Reff  R / 1  C E / CQ                                     (18-b)

while the effective capacitance in this case is the conventional electrostatic capacitance C E .
In addition to this equivalent distributed circuit, two additional contact resistances should
be included at the two ends of the equivalent circuit of the carbon nanotube transmission
line. The value of this contact resistance is given by Rc  h / 8 e 2 .
By comparing these values, it can be noted that the kinetic inductance is much larger than
the magnetostatic inductance of transmission line section where the ratio of
 Lk / LM  4  10 3 . This means that the kinetic inductance has the dominant inductive effect
on the equivalent distributed circuit. On the other hand, the quantum capacitance is nearly
of the same order of the electrostatic capacitance of the transmission line section. This
property has two main effects on electromagnetic wave propagation along the carbon
nanotube transmission line; slow wave propagation and high characteristic impedance. The
complex propagation constant, phase velocity and characteristic impedance in this case are
given by:

                                                                       
                                                 j  jCE Reff  jLeff                                       (19-a)




                                                                                              
                                                                          
             lim            vp     lim                                                             1 / Leff C E   (19-2)
            R  L                R  L
                                                        C ELeff   4C EL2   2C E Reff
             eff      eff           eff      eff
                                                     1 2                2          2 2
                                                                          eff
                                                     2

                                                    Zc     Reff  jLeff  / jCE                                 (19-c)

The phase velocity in this case is nearly of the same order of Fermi velocity vF which is
nearly two-order less than free space light speed. This means that the wavelength along
carbon nanotube transmission line is nearly two-order less than conventional transmission




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480                                                                  Carbon Nanotubes Applications on Electron Devices

lines. This property has a significant importance in RF applications since the dimensions of
passive circuits like filters, couplers and power dividers are comparable with operating

characteristic impedance of order 12.5 k . Recently, parallel carbon nanotubes and carbon
wave length. However, these additional inductance and capacitance introduce high

nanotube bundle have been introduced to overcome this disadvantage. The effective
distributed elements of the equivalent circuit for a carbon nanotube bundle are simply the
parallel combination of the circuit for a single carbon nanotube. Thus, both the effective
inductance and resistance are divided by N and the effective quantum capacitance is
multiplied by a factor N where N is the number of the nanotubes in the bundle. Different
experimental results have shown that using parallel carbon nanotube decreases both the DC
and RF impedance. However, it increases the wave velocity and subsequently the wave
length. Thus, a compromise between the required wave velocity and characteristic
impedance should be considered to select the appropriate number of carbon nanotubes in

operating frequencies less than electron relaxation frequency where   Reff / Leff   . Thus
the bundle. On the other hand, the attenuation coefficient has a significant effect at the


at lower frequency band below relaxation frequency, carbon nanotube presents a good
candidate for an absorbing structure more than a guiding structure.

4. Finite difference analysis of coupled Maxwell-Schrödinger equations
In this section we present another approach which is useful to study electromagnetic
interaction with nanodevices like carbon nanotubes. This approach is based on coupling
Schrödinger equation which describes the motion of charged particles along the nanodevice
with Maxwell’s equations which describe the electromagnetic waves in the region of the
nanodevice. Both Schrödinger and Maxwell’s equations can be presented as partial
differential equations which can be solved numerically by using finite difference scheme
based on specific sources and boundary conditions. This approach is quite useful to study
electromagnetic field interaction with short nanotubes of length less than 100 nm where the
electron transport is nearly ballistic.
The quantum motion of electrons along a nanodevice can be presented by Schrödinger
equation as follows:

                                        h  ( r , t )                     
                                                         2  2  V  r   ( r , t )
                                                            h2
                                       2   t           8 m              
                                                                           
                                   j                                                                             (20)


where  ( r , t ) is the complex state variable of the electron on the nanodevice and V  r  is the
static potential along this nanodevice. The key difference between different nanodevices lies
in this static potential. In the presence of time-varying electromagnetic fields, Schrödinger
equation is modified as follows:

                      h  ( r , t )    1  h2 2
                                                  ( r, t )  j           A( r , t ) ( r , t )
                     2                2 m  4 2                   2
                                                                     he
                          t               
                 j

                                                                                                             
                                             j    A( r , t )   ( r , t )  e 2 |A( r , t )|2  ( r , t ) 
                                                2
                                                he
                                                                                                             
                                                                                                                 (21)

                                            e ( r , t ) ( r , t )  V  r  ( r , t )




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where A( r , t ) is the applied magnetic vector potential,  ( r , t ) is the applied scalar electric
potential and e here represents the absolute value of the electron charge. The resulting
quantum current density along the nanodevice can be presented in terms of this electron
state variable as follows:


                J( r , t ) 
                               e2
                               m
                                  | ( r , t )|2 A  j
                                                       4 m
                                                        he
                                                                  
                                                             * ( r , t ) ( r , t )   ( r , t ) * ( r , t )        (22)

On the other hand, electromagnetic fields are related by Maxwell’s equations as follows:

                                                                      E( r , t )
                                                H( r , t )                     J( r , t )
                                                                        t
                                                                                                                        (23-a)


                                                                              H( r , t )
                                                    E( r , t )   
                                                                                t
                                                                                                                        (23-b)

where the electric and magnetic field components can be presented in terms of the scalar
electric potential and vector magnetic potential as:

                                                                                 A( r , t )
                                               E( r , t )   ( r , t ) 
                                                                                   t
                                                                                                                        (24-a)


                                                   H( r , t )              A( r , t )
                                                                  
                                                                      1
                                                                                                                        (24-b)

The scalar electric potential and the magnetic vector potential are related by Lorentz
condition:

                                                   ( r , t )
                                                                     A( r , t )
                                                                  
                                                                   1
                                                    t
                                                                                                                        (25-a)

and the magnetic vector potential is related to the current density by non-homogenous wave
equation:

                                            2 A( r , t ) 1 2
                                                              A( r , t )  J( r , t )
                                                                          
                                                                            1
                                               t 2
                                                                                                                        (25-b)

By solving Equations (21)-(25), one can obtain electromagnetic interaction with the proposed
nanodevice. Numerical solution of these coupled partial differential equations can be obtained
by using finite difference method. This method is based on approximating the differential

mesh in the space (ix , jy , kz) at discrete steps in time (nt ) . To simplify the problem, the
operator by a difference operator to obtain the required quantities on the nodes of a discrete

complex state variable of Equation (21) is divided into real and imaginary parts as follows:

                                                ( r , t )   R ( r , t )  j I ( r , t )                               (26)

Thus, Equation (21) can be reformulated as two coupled partial differential equations of
pure real variables:




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482                                                                   Carbon Nanotubes Applications on Electron Devices

                       R ( r , t )    1  h 2
                                                I ( r , t )  e  A( r , t ) R ( r , t )
                         t            2 m  2
                                                                               2 2                                
                                             2 eA( r , t )   R ( r , t )        e |A( r , t )|2  I ( r , t ) 
                                                                                                                   
                                                                                                                          (27-a)
                                                                                h
                                           2                             2                      
                                             e ( r , t ) I ( r , t )     V  r  I ( r , t ) 
                                           h                               h                      

                       I ( r , t )    1  h 2
                                                R ( r , t )  e  A( r , t ) I ( r , t )
                         t            2 m  2
                                                                                2 2                               
                                              2 eA( r , t )   I ( r , t )       e |A( r , t )|2  R ( r , t ) 
                                                                                                                   
                                                                                                                          (27-b)
                                                                                 h
                                          2                             2                      
                                            e ( r , t ) R ( r , t )     V  r  R ( r , t ) 
                                          h                               h                      
For the case of a carbon nanotube where the length of the nanotube is usually much larger

nanotube. In this case the  operator in the above equation can be simply replaced by
than its diameter, the problem can be presented as 1-D problem along the length of the

  / z a z .

time steps n and n  1 / 2 . Thus by discretizing Equation (25-b) one can obtain the update
The scalar electric potential and magnetic vector potentials are assumed to be known at the

equation of the magnetic vector potential at time step n  1 as follows:

      Az  1 ( k )  2 Az  1/2 ( k )  Az ( k )
       n                n                n



                                2  z                                                      z
                  
                       t / 2 2 An  1/2 ( k  1)  2 An  1/2 ( k )  An 1/2 ( k  1)   t / 2 2 J n  1/2 ( k )
                        z 
                                                                                                                            (28)
                                                         z                z



Based on the result of Equation (28) and the previously stored electric potential and vector

time step n  1 / 2 by discretizing Equation (24-a)
potentials, one can obtain the update equation for the electric field along the nanotube at the


                                               n  1/2 ( k  1)   n  1/2 ( k  1)         Az  1 ( k )  Az ( k )
                         Ez  1/2 ( k )                                                 
                                                                                               n              n

                                                               2 z                                     t
                          n
                                                                                                                            (29)

The update equations of the electric fields at the time step n + 1/2 in the remaining domain
outside the nanotube can be obtained by using conventional FDTD formulation for the
differential form of Ampere’s law, Equation (23-a),

                                                                                  t
                                      En  1/2 ( r , t )  En  1/2 ( r , t )           Hn ( r , t )
                                                                                  
                                                                                                                            (30)

It should be noted that we assumed here the present current density is limited to the
nanotube structure only which is already included in Equation (29). Thus Equation (30) does
not include an electric current term. Similarly, the update equations of the magnetic fields at
the time step n + 1 are obtained by using conventional FDTD formulation for the differential
form of Faraday’s law, Equation (23-b),




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                                                                                  t
                                             Hn  1 ( r , t )  H n ( r , t )           En  1/2 ( r , t )
                                                                                  
                                                                                                                              (31)

The update equations of the scalar electric potential function at the time step n+1 can be
obtained by discretizing Equation (25-a) as follows:

                                n1 ( k )   n ( k ) 
                                                             1 t
                                                             2 z
                                                                        A  n  1/2
                                                                            z       (k      1)  Az  1/2 ( k  1)
                                                                                                   n
                                                                                                                             (32)

To obtain the above quantities at the following time step, it is required to update the
potential functions and the current density to the time step n+3/2. To do this it is required to
use Schrödinger equation to update the state variable and consequently the current density
and the potential functions. Assuming that the state variables are known at time steps n and
n+1/2, one can obtain the temporal update equation of the real and imaginary parts of the
complex state function at the time step n+1 in the following forms:

                                       ht  In  1/2 ( k  1)  2 In  1/2 ( k )   In  1/2 ( k  1)
       R 1 ( k )   R ( k ) 
                                       4m                          z 2
        n              n



                        et Az  1/2 ( k  1)  Az  1/2 ( k  1) n  1/2
                                                                     R
                             n                        n

                                               2 z
                                                                              (k)

                        et n  1/2  R 1/2 ( k  1)   R 1/2 ( k  1) t 2 n  1/2
                                                                                                                         
                        2m                                                                                           (33-a)
                                                                                                 (k)  I
                                                                                                         n  1/2
                                              n                     n                                 2

                                                               2 z
                            Az      (k)                                                       e Az               (k)

                        2t n  1/2                            2t
                         m                                                             mh
                                                                      V  k  In  1/2 ( k ) 
                                                                                              
                           e          ( k ) In  1/2 ( k ) 
                        h                                         h                          

                                       ht  R 1/2 ( k  1)  2 R 1/2 ( k )   R 1/2 ( k  1)
        In  1 ( k )   In ( k ) 
                                                                  z 
                                             n                    n                n

                                       4m                              2


                         et Az  1/2 ( k  1)  Az  1/2 ( k  1) n  1/2
                                                                   I
                               n                     n

                                               2 z
                                                                             (k)

                         et n  1/2  In  1/2 ( k  1)   In  1/2 ( k  1) t 2 n  1/2
                                                                                                                         
                         2m                                                                                      (33-b)
                                                                                             (k)  R
                                                                                                     n  1/2
                                                                                                  2

                                                             2 z
                             Az      (k)                                                  e Az               (k)

                          2t n  1/2                         2t
                          m                                                        mh
                                                                     V  k  R 1/2 ( k ) 
                                                                                           
                              e        ( k ) R 1/2 ( k ) 
                          h                                                               
                                                  n                           n
                                                                  h

Based on this complex state variable, one can obtain the current density at the time step
n+1by discretizing Equation (22) as follows:

                      e2  n 1                     2  n1  he  n  1  R 1 ( k  1)   R 1 ( k  1) 
       Jz 1(k)           R ( k )   I ( k )  Az        I ( k )                               
                                            n1
                                                                            n               n

                                                            
                                                            2 m                                         
                                       2

                      m                                                              2 z
        n

                                                                                                          
          he  n  1  In  1 ( k  1)   In  1 ( k  1) 
                                                                                                                              (34)
              R ( k )                                   
         2 m 
                                    2 z                  
                                                           
At this point we have  n  1 ( k ) , Az  1 ( k ) ,  R 1 ( k ) ,  In  1 ( k ) and J z  1 ( k ) in addition to the
                                       n               n                                 n

previously calculated quantities. By repeating the same steps from Equations (32) to (34) and




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484                                                   Carbon Nanotubes Applications on Electron Devices

Equation (28), one can obtain state parameters, current density and potential functions at
n+3/2 time step. Thus we obtain scalar electric potential and magnetic vector potentials at
the time steps n+1 and n+3/2. Then by repeating the same steps of Equations (28) to (31) one
can obtain the electric and magnetic fields in the following time step.
The above analysis represents the core of the finite difference time domain formulation of
the coupled Maxwell-Schrödinger equations for solving electromagnetic coupling with
nanodevice with emphasis on simple linear nanodevices like carbon nanotube. It should be
noted that there are many other problems in this method which require more investigation
like absorbing boundary conditions for Schrödinger equation, stability and dispersion.
(Pierantoni et al.; 2008, Pierantoni et al.; 2008 & Ahmed et al. 2010)

5. Surface wave propagation along carbon nanotubes
Carbon nanotubes can also be considered as cylindrical guiding structures of finite
conductivity. This section shows how to determine the complex propagation constant of this
guiding structure based on its macroscopic conducting properties. Since we are mainly
concerned with the longitudinal conductivity, the propagating wave along the nanotube
would be mainly TM wave. In this case the total field can be represented in terms of the
axial TM Hetezian potential e as follows:

                                         E       e   k0  e
                                                             2
                                                                                                (35-a)

                                            H  j o   e
                                            
                                                                                                (35-b)

where the TM Hertezian potential is determined by solving the wave equation

                                            2  e  k0  e  0
                                                      2
                                                                                                  (36)


Bessel functions. The field inside the cylinder is finite in the range 0    r where r is the
For a cylindrical configuration, the general solution of wave equation can be presented as


function of first kind. On the other hand, the field outside the cylinder is finite at  = r and is
radius of the proposed cylinder. Thus the field in this region is represented by Bessel

exponentially decaying for  = r. Thus, the field in this region is represented by Hankel
function of second kind. Hence, the general solution of the TM Hertizian potential in carbon
nanotube (single or a circular bundle) can be represented as:


                                    J ( )Hn ( r )  j z  jn
                                                                          r
                            e  Aaz  n                 e
                                                  (2)


                                      J n ( r )Hn ( )
                                                        
                                                                            r
                                                              e                                   (37)
                                                  (2)



By using this Hertezian potential in Equation (35) and applying the boundary condition

                           J z   zzEz (r )   lim  H (r   )  H (r   )
                                                                               
                                                0
                                                                                                  (38)


one can obtain the dispersion equation for surface wave propagation as follows:




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Carbon Nanotubes in Passive RF Applications                                               485


                       
                        J n ( r )Hn ( r )              where Im( )  0
                           2


                                                 zzZ0 k0r
                                                     2
                       k0 
                                     (2)
                                                                                          (39)

The longitudinal propagation constant is given by:

                                    k0   2 where Im( )  0
                                       2
                                                                                          (40)

It would be useful here to study the limit of the above dispersion equation for the zero-order
mode at small argument limit. In this case the Bessel function combination of the right hand
side can be approximated as:

                                                                                   
                       J 0 ( r )H(2) ( r )   1  j ln  r / 2)  0.577215   
                                                                                 
                                                      
                                                      2
                                                                                   
                                  0                                                       (41)

Unlike Bessel function, the logarithmic function in the right hand side of Equation (41) can
be represented by a slowly convergent series for small argument. However, for gigahertz
frequency band, the average value of this logarithmic function is nearly around minus ten.
Thus, an approximate value of the zero order mode complex surface wave propagation
coefficient along a carbon nanotube is given by:


                                   k0 1 
                                                zzZ0 k0r  / 2  j10 
                                                           1
                                                                                          (42)


It can be noted that, by increasing the longitudinal conductivity of the tube the surface wave
propagation constant approaches the free space propagation constant.




bundle is composed of armchair carbon nanotubes with lattice parameters m  n  40 .
Fig. 6. Surface wave propagation on carbon nanotube bundles of different values of N. The

(Attiya 2009)
Figure 6 shows complex wave propagation constant of TM surface waves along a carbon
nanotube circular bundle for different values of N where N is the number of carbon nanotubes
in the bundle. The present results are based on armchair carbon nanotubes of m = n = 40. For




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486                                                           Carbon Nanotubes Applications on Electron Devices

this configuration, the radius of the nanotube is rc = 2.7nm. For a closely backed bundle
composed of N carbon nanotubes arranged in a single circular shell the radius of the bundle is

                                                     r  2 Nrc /                                                     (43-a)

In this case the effective axial surface conductivity of this shell can be approximated by

                                                   zz  N zz rc / r
                                                    b       cn
                                                                                                                      (43-b)

It can be noted that the attenuation coefficient increases by decreasing the operating
frequency as shown in Figure 6. The effect of this attenuation coefficient is negligible in the
frequency range from 100 to 1000 GHz. On the other hand, this attenuation coefficient has a
significant effect in the frequency band below 100 GHz.

6. Carbon nanotube antenna
Carbon nanotube can also be a good candidate for antenna structures. Slow wave property
of electromagnetic propagation along carbon nanotube is expected to play an important role
in reducing the size of resonant carbon nanotube antenna. From the mathematical point of
view, carbon nanotube antennas can be treated as an antenna composed of finite-conducting
cylinders. In this case the dynamic conductivity derived in Section 2 is used to include the
electrical properties of carbon nanotube in the mathematical modeling of the corresponding
antenna structure. This electromagnetic formulation can be presented in any form like
integral equation, finite difference or finite element. However, for simple wire antenna
configuration, electric field integral equation method may be the most appropriate method.
Thus we would focus on this method in the following part of this section.
For the case of a simple dipole antenna oriented along the z axis, the relation between the
excitation field and the current distribution can be presented by Hallen’s integral equation
as follows:

               jk0  z  z2  r 2                          
                                                                                            4 0
          
             e                                e  jko|z  z| I ( z)dz  C cos k0 z  j
                                                             
                                                                                                   sin k0 |z  z0 |
         L

                                                               
          L   z  z   r
                                        Z0 zzr
                                          1
                                                               
                                                                                                                        (44)
                                                              
                           2        2                                                        2 k0

where r is the radius of the dipole ( r  rc for single nanotube) and  zz is the effective
surface conductivity of the dipole, z0 is the location of the feeding point, C is a constant that
would be determined to satisfy the current vanishing at the edges of the dipole and L is the
half length of the dipole. The effect of the carbon nanotube in this integral equation lies in
surface conductivity and the radius of the nanotube. This formulation can be used to
simulate a dipole antenna composed of single carbon nanotube or a bundle of carbon
nanotubes. In the case of a carbon nanotube bundle, the corresponding radius and surface
conductivity are obtained by Equation (43).
This integral equation can be solved numerically by using method of moments to find out
the current distribution and subsequently the input impedance, radiation pattern, radiation
efficiency and other antenna parameters. This method is based on expanding the unknown
current distribution as a finite series of known basis functions of unknown amplitudes. To
determine these amplitudes, Equation (44) is weighted by a set of weighting functions




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Carbon Nanotubes in Passive RF Applications                                               487

which equals the number of the unknown amplitudes. Thus, Equation (44) is converted into
a system of equations which is solved to obtain the unknown amplitudes of the current
distribution functions. Details of solving this integral equation numerically by using method
of moments can be found in (Elliot 2003).
Figure 7 shows the complex input impedance of a dipole of a carbon nanotube for two cases

armchair carbon nanotubes of m = n = 40. The length of the dipole is assumed to be 30 m
as functions of the operating frequency. The present results are based on conducting

and 1 mm for the first and second case respectively. It can be noted that the first case has a

Figure 8-a shows the current distribution along the 10 mdipole antenna at its first
first resonance at nearly 160 GHz. However, the second case does not have any resonance.

resonance frequency due to a unity voltage source at its feeding point. It can be quite clear
that the dipole in this case corresponds to a half-guided-wave length dipole. It can also be
noted that the length of the first resonant carbon nanotube antenna is nearly 0.0107 times the
length of conventional half-wave length dipole at this frequency. This represents an
important feature of carbon nanotube antenna. It can also be noted the corresponding input
impedance in this case is nearly 11 kΩ which is much greater than the conventional input
impedance of a half-wave dipole which is nearly of order 75Ω. On the other hand, the
current distribution along the 1 mm dipole antenna has different properties as shown in
Figure 8-b, where the real and imaginary parts of the current are concentrated around the
center of the dipole and highly damped at its ends. It can be concluded from this result that
the problem of carbon nanotube antenna at different frequency ranges cannot be directly
obtained by simple scaling as the case in perfect electric conductor.

values of N. The length of the dipole is taken to be 30 m and 3000 m. It can be noted that
Figure 9 shows the input impedance of a dipole of a carbon nanotube bundle for different

the 30 m dipole has a first resonance, that corresponds to a half-guided-wave length dipole
for N = 8, at 280 GHz. The resonance impedance in this case is nearly 2100 Ohms. By




(a) The length of the dipole is 10 m         (b) The length of the dipole is 1 mm
Fig. 7. Input impedance for a carbon nanotube antenna. The carbon nanotube is armchair of
lattice parameters m=n=40.




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488                                             Carbon Nanotubes Applications on Electron Devices

comparing this length with free space half-wave at this frequency it can be noted that this
carbon nanotube antenna has a reduction scale factor of nearly 0.056 compared with
conventional half-wave length dipole. Increasing the number of nanotubes in the bundle
decreases the total surface impedance of the dipole. This has two effects, increasing the
resonance frequency for a specific length and decreasing the resonance impedance as shown
in the case where N is increased to twenty. In this case the first resonance frequency is 404
GHz and the resonance impedance is 840 Ohms. The scale reduction factor in this case is
nearly 0.081. For a hundred nanotube bundle of the same length, the resonance frequency
would be 740 GHz and the resonance impedance would be 174 Ohms. In this case the scale
reduction factor is nearly 0.15. It can be noted that by increasing the number of carbon

the input impedance is decreased. On the other hand, the 3000 m dipole does not introduce
nanotubes in the bundle, both resonance frequency and reduction factor are increased while

resonance behavior for any value of N as it is shown in Figure 9-b. This result is quite similar
to the results of a single carbon nanotube antenna shown in Figure 7.




(a) The length of the dipole is 10 m and      (b) The length of the dipole is 1 mm and
the operating frequency is 160 GHz             the operating frequency is 10 GHz
Fig. 8. Current distributions along carbon nanotube dipole antennas due to a unity voltage
source.




(a) L = 30 m                                 (b) L = 3000m
Fig. 9. Input impedance of bundle dipole. The bundle is composed of armchair carbon
nanotubes with lattice parameters m = n = 40. Numbers of nanotubes in the bundles are
N=8, N=20 and N=100 (Attiya 2009)




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It is noted that the inverse reduction factors of the resonant bundle dipoles equals nearly the
ratio of the surface wave propagation constant with respect to the free space wave number
shown in Figure 6 for the same bundles. These results show the relation between the
resonant dipole length and the surface wave velocity on its arms. On the other hand, by
studying the surface wave complex wave propagation constant, it can be noted that the
attenuation coefficient increases by decreasing the operating frequency as shown in Figure
6. The effect of this attenuation coefficient is negligible in the frequency range from 100 to
1000 GHz. Thus, the main behavior of the input impedance of the dipole antenna is nearly
the same of traditional dipole antenna with taking into account scaling reduction factor due
to the slow surface wave velocity. However, in the band from 1 to 10 GHz, the wave
propagating on the arms of the dipole is attenuated. Thus, the reflected wave does not add
completely at the feeding point which means the inductive effect due to the delayed
reflected signal does not compensate completely the capacitive effect of the dipole arms.
This explains the capacitive behavior of CNT dipoles in Figures 7-b and 9-b. In this case, the
wave propagating on the arms of the dipole is highly attenuated, such that the active part of
the dipole is much smaller than the physical length of the dipole itself. Thus, the dipole
would always be a short dipole in this case and it is not resonant in any case. This result
shows that the advantage of size reduction combined with surface wave propagation can be
used only in high frequency bands above 100 GHz.

7. Carbon nanotubes in passive RF circuits
Recent advances in carbon nanotubes make them competitive elements in many RF
applications. New fabrication techniques can be used to synthesize and electrically contact
single carbon nanotube up to nearly 1 cm (Li et al. 2007). In addition using solubilized
carbon nanotube and dielectrophoresis can be used to accumulate hundreds to thousands of
carbon nanotubes in parallel (Rutherglen et al. 2008). These advances in fabrication
techniques open the door for more research on different configurations of carbon nanotubes
which are believed to be visible in near future.
On the other hand, metallic single-wall carbon nanotube transmission line shows an
important advantage of slow electromagnetic wave propagation compared with free space
wave velocity. This wave velocity reduction is due to the additional kinetic inductance and
quantum capacitance in the equivalent circuit model of the nanotube transmission line
circuit. In this case the wave velocity along the single-wall carbon nanotube transmission
line has the same order of Fermi velocity which is nearly 8 x 105 m/s. This means that the
wave velocity in this case is nearly two-order less than free space wave velocity. Thus the
wavelength along the single-wall carbon nanotube transmission line is nearly two-order
smaller than free space wave length. This property is quite useful to reduce the physical
dimensions of microwave circuits to be in hundred-micrometer scale instead of centimeter

impedance of order 12.5 k . In addition, the single-wall carbon nanotube transmission line
scale. However, these additional inductance and capacitance introduce high characteristic

has an intrinsic resistance of 6.5 k and Ohmic contact resistance of order 20 k which
introduce high attenuation coefficient. These parameters make carbon nanotube
transmission line is not suitable for microwave applications from the point of view of
characteristic impedance and attenuation.
Recently, parallel carbon nanotube and carbon nanotube bundle have been introduced to
overcome these disadvantage (Attiya & Kanhal, 2009). Different experiments introduced by




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490                                           Carbon Nanotubes Applications on Electron Devices

different authors have shown that using parallel carbon nanotube decreases both the DC
and RF impedance; for example a bundle composed of ten parallel single wall carbon
nanotube shows a DC impedance around 750 while hundreds of parallel single wall
carbon nanotubes show an AC impedance from 60 to 40 in the frequency range from few
MHz to 20 GHz (Rutherglen et al. 2008). These experimental results introduced the
possibility of using parallel single wall carbon nanotubes to obtain nearly matched
transmission line sections with low wave velocity. Parallel carbon nanotube transmission
line sections can be used to replace traditional printed transmission line sections in
microwave circuits to have a significant reduction in the total size of these circuits.
However, one should consider the lossy and mismatch effects of parallel carbon nanotube
transmission line. Further theoretical and experimental investigations are still required to
study the possibility of using parallel carbon nanotube transmission lines in passive
microwave circuits like hybrid couples, power dividers, filters… etc.
Other RF and microwave applications like switches, filters and resonators can also be
obtained by using electromechanical propertied of carbon nanotubes. (Demoustier et al.
2008) introduced an RF nanoswitch based on vertically aligned carbon nanotubes. It consists
of carbon nanotube perpendicular to the substrate. Two different architectures are proposed
for this carbon nanotube switch; series-based switch using ohmic contact between carbon
nanotubes and a capacitive-based switch implemented in shunt configuration. RF ohmic
switch is designed by implementing carbon nanotubes in two sides of a coplanar waveguide
discontinuity as shown on Figure 10-a. By applying dc voltage on the two sides of the
coplanar waveguide discontinuity, an electrostatic force is introduced between the two arms
of the carbon nanotube switch. This electrostatic closes the switch and the RF signal is
transmitted across the coplanar waveguide as shown in Figure 10-b. On the other hand,
shunt switch is based on two nanotube capacitive contacts between the inner line and the
two sides of the ground planes of the coplanar waveguide as shown in Figure 11-a. By
applying dc voltage between the inner and outer sides of the coplanar waveguide, the
electrostatic field introduces a short circuit between the inner and the outer sides as shown
in Figure 11-b. This short circuit reflects the propagating wave along the coplanar
waveguide which corresponds to switching off the RF signal. To achieve the expected
performance in the required operating frequency band, two inductive sections are added
along the coplanar waveguide in series with nanotube to perform a series LC resonance at
the center of operating frequency band. This resonant LC circuit introduces higher isolation
level in the switch isolation state.




             (a) Isolation state                         (b) Transmission state
Fig. 10. Architecture of Carbon nanotube ohmic switch (Demoustier 2008)




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Carbon Nanotubes in Passive RF Applications                                               491




                                                                                            .
           (a) Transmission state                                (b) Isolation state
Fig. 11. Architecture of Carbon nanotube shunt capacitive switch (Demoustier 2008)
Another application for carbon nanotubes in passive RF circuits based on their
electromechanical properties is the microwave resonator and filter (Dragoman 2005). It is
found that carbon nanotube has a mechanical resonance in the frequency range from 1 to 3
GHz with quality factor of 1000. The basic theory of carbon nanotube filter is based on
coupling the electromagnetic fields of the incident signal to a perpendicular array of carbon
nanotubes. This can be obtained by inserting this array of carbon nanotubes inside a
coplanar waveguide as shown in Figure 13. In this case the coplanar waveguide transmits
only the signals which are resonant with carbon nanotube array. To introduce the coupling
between the incident electromagnetic wave and the carbon nanotube array, it is required to
produce electric charges on the carbon nanotube. These electric charges are obtained by
applying a dc electric field parallel to the direction of electromagnetic wave propagation and
orthogonal on carbon nanotubes as shown in Figure 13. The presence of these electric
charges introduces Coulomb forces between the carbon nanotubes and the electric field of
the incident wave. The resonance of this filter is controlled by the value of the applied dc
voltage. The vibration of the excited tubes located near the input electrode is propagating
along the entire array like an acoustic excitation.




Fig. 13. Architecture of an RF bandpass filter based on a carbon nanotube array.




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492                                                         Carbon Nanotubes Applications on Electron Devices

8. Carbon nanotubes composites for RF absorbing and shielding
applications
From the above discussions about the possibility of using carbon nanotubes in microwave
circuits it can be concluded that the high attenuation coefficient of wave propagation along
carbon nanotube represents a main limiting factor in these application which may be
overcome by using bundle of carbon nanotubes. However, this attenuation property is quite
important in other microwave applications like absorbing and shielding. For these
applications it is not required to arrange the carbon nanotubes in a bundle like the cases of
antennas, transmission lines and interconnects. However, carbon nanotubes in these cases
are mixed with other materials in a random form.
The effective dielectric constant of a dielectric mixture including conducting particles can be
formulated by using Maxwell-Granett mixing rule as follows (Koledintseva 2009):


                                            fi  i   b  
                                                                          b
                                                                  b  N ik   i   b 
                                                              3
                                          1

                          eff   b                       k 1
                                          3

                                                f i  i   b  
                                                                                                        (45)
                                         1
                                                                       b  N ik   i   b 
                                                                   3
                                              1                               N ik
                                              3                  k 1

where  b is the relative permittivity of the base dielectric,  i is the permittivity of the

depolarization factor of the inclusions and the index k  1, 2, 3 corresponds to x , y and z in
inclusion particles, f i is the volume fraction occupied by the inclusion, N ik is the

Cartesian coordinates. The depolarization factors for an ellipsoid inclusion of radii a,b and c
in x , y and z directions are given by:



                                       2  (s  a) ( s  a)(s  b )(s  c )
                                           
                             Ni1 
                                      abc                 1
                                                                            ds                        (46-a)
                                          0




                                       2  (s  b ) (s  a)(s  b )(s  c )
                                           
                             Ni 2 
                                      abc                 1
                                                                            ds                        (46-b)
                                          0




                                       2  (s  c ) (s  a)(s  b )(s  c )
                                           
                             Ni 3 
                                      abc                 1
                                                                            ds                        (46-c)
                                          0

For the case of a carbon nanotube inclusion a = b= rc and c corresponds to the half-length of
the carbon nanotube. To simplify the formulation it is assumed that all nanotube inclusions
are directed parallel to the z axis. In this case the depolarization effect of the inclusions
would be mainly dominating in the z direction while the dielectric constant in transverse

carbon nanotube inclusion is defined as  i  j 3D /  where the equivalent 3D conductivity
directions would be the same as the base dielectric. In this case the dielectric constant of the



 3D   zz / 2 rc (Mikki & Kishk 2009). Different experimental researches have been
of carbon nanotube is related to the surface dynamic conductivity of carbon nanotube as

introduced to study the electrical prosperities of carbon nanotube composites. We
demonstrate two specific cases in this following part of this section as examples for these
carbon nanotube composites.




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Carbon Nanotubes in Passive RF Applications                                                493

In microwave hyperthermia applications, it is required to an increase electromagnetic power
dissipation in the specific region that is required to be more heated than other regions.
Carbon nanotubes are found to be a good candidate for this application. To verify the
applicability of carbon nanotube in this application (Mashal et al. 2010) demonstrated an
experiment based on mixing tissue mimicking materials with carbon nanotubes and
measuring their electrical properties and their heating response to incident electromagnetic
wave. These tissue mimicking materials are constructed from oil-in-gelatin dispersions. The
dielectric properties of these materials are customized to mimic the properties of a variety of
human soft tissues by controlling the concentrations of gelatin, safflower oil, kerosene, and
preservatives. The carbon nanotubes used in their experiments were 1-2 nm in diameter and
5-30 μm in length, and were composed of mainly single wall carbon nanotubes. Figure 14
shows the measured relative permittivity and effective conductivity of the corresponding
composite for different concentrations of carbon nanotubes. It can be noted that both the
permittivity and effective conductivity of the tissue mimic materials increase by increasing
concentration of the carbon nanotubes.
The electromagnetic heating responses of this tissue mimic composites with carbon

an inner cross section 72 mm  34 mm and applying a 3-GHz CW signal of power 1 Wt. The
nanotube were examined by inserting a sample inside a WR-284 rectangular wave guide of

source generator is turned on for 3 minutes to heat the sample and turned off for 5 minutes
to cool the sample. Figure 15 shows the measured heating responses for different values of
carbon nanotube concentrations. It can be noted that the maximum temperature of the tissue
mimic mixture increases by increasing the concentration of carbon nanotubes.
It can be concluded from these results that low concentrations of carbon nanotubes
significantly impact the dielectric properties and heating response of tissue mimicking
materials. For example, at 3 GHz, carbon nanotubes concentrations as small as 0.22% by
weight increased the relative permittivity of the tissue mimicking material by 37% and the
effective conductivity by 81%. This concentration of carbon nanotubes led to an average
steady-state temperature rise that was 6 oC higher than the rise observed in the tissue
mimicking material without carbon nanotubes. These results suggest that carbon nanotubes
may enhance contrast for microwave imaging and facilitate selective microwave heating for
treatment of breast cancer (Mashal et al. 2010).




         (a) Relative permittivity                    (b) effective conductivity
Fig. 14. Electrical properties of tissue-mimicking mixtures with varying concentrations of
single wall carbon nanotubes measured from 0.6 to 20 GHz. (Mashal et al. 2010)




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494                                            Carbon Nanotubes Applications on Electron Devices




Fig. 15. Microwave heating response of tissue mimic materials with various concentrations
of carbon nanotubes. Each curve shows the temperature profile of a sample that was heated
via 3-GHz microwave illumination for 3 min and allowed to cool for 5 min. (Mashal et al.
2010)
Another important application of carbon nanotube composites is optical-transparent
electromagnetic shielding composite film. Thin films of conducting single wall carbon
nanotube of a thicknesses less than 300 nm on polyethylene terephthalate substrates are
good candidate for this application. (Xu et al. 2007) introduced an experimental study to
characterize the shielding properties of these composite films. Their study is based on
measuring the reflection coefficient of a coaxial annular ring resonator placed above the
carbon nanotube layer by using a vector network analyzer. Based on this reflection
coefficient they obtained the equivalent impedance of the composite film. This impedance
includes the impedance of the carbon nanotube film and the impedance of the holding
substrate. To extract the impedance of the substrate, they measured the reflection coefficient
of the coaxial annular resonator on the substrate only. After extracting the impedance of the
substrate, they obtained the impedance of the carbon nanotube layer. This impedance is
used to determine the equivalent complex conductivity of the carbon nanotube layer. Then
the problem is treated as a two-layered structure of specific values of permitivities and
conductivities to determine the transmission coefficient of this structure.
Figure 16 shows the shielding effectiveness in dB as a function of frequency for different
thicknesses of carbon nanotube films. It is noted that this shielding effectiveness is
proportional to log (1/). For the 10 nm film, the shielding effectiveness varies from 43 to 28
dB in the range of 10 MHz–30 GHz. The dependence of the shielding effectiveness on the
thickness of the carbon nanotube layer t0 is nearly proportional to log (t0). On the other
hand, Figure 17 shows the optical transmission coefficient at wavelength of 550 nm as a
function of the thickness of the carbon nanotube film. For a 30 nm thickness film, the optical
transmittance is about 80% and the shielding effectiveness are 33 dB at 10 GHz, 36 dB at 1
GHz, and 46 dB at 10 MHz. This shielding effectiveness of carbon nanotube films satisfies
requirements for commercial applications like cell phones which require approximately 20
dB shielding effect. For high shielding requirements such as for magnetic resonant imaging
window where 60 dB shielding effectiveness is required, carbon nanotube films still need to
be improved.




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Carbon Nanotubes in Passive RF Applications                                               495




Fig. 16. Microwave shielding effectiveness of a carbon nanotube film for different values of
thicknesses. (Xu et al. 2007)




Fig. 17. Optical transmission coefficient at wavelength of 550 nm as a function of the
thickness of carbon nanotube film. (Xu et al. 2007)

9. Conclusion
This chapter introduced different techniques for studying interaction of high frequency
electromagnetic fields with carbon nanotubes. Boltzman kinetic equation is used to
introduce an equivalent surface conductivity and electron fluid model is used to introduce
an equivalent circuit model for carbon nanotube transmission line. Another model is
introduced based on coupling Maxwell’s equation with Schrodinger equation. Finite
difference time domain is discussed as an efficient numerical technique for solving coupled
Maxwell-Schrodinger equations to obtain a full wave analysis for electromagnetic
interaction with carbon nanotubes.
These models show that carbon nanotubes are characterized by high inductive effect due to
the additional kinetic inductance. This high inductive effect reduces wave velocity along
carbon nanotubes and increases its corresponding characteristic impedance. Reduction of
wave velocity has a significant importance in reducing the size of RF components, passive
circuits and antenna structures. On the other hand, parallel carbon nanotubes can be used to
reduce the characteristic impedance. Analytical analysis of surface wave propagation along
circular carbon nanotube bundle is discussed based on the equivalent surface conductivity.
Resulting complex wave propagation constant along carbon nanotube bundles shows slow
wave propagation which is consistent with the transmission line model introduced by




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496                                            Carbon Nanotubes Applications on Electron Devices

electron fluid model. In addition, attenuation coefficient is found to be increased by
decreasing the operating frequency. Carbon nanotubes are also found to be a good
candidate for dipole antennas at operating frequencies above 100 GHz. At lower frequencies
the high attenuation coefficient of wave propagation along the carbon nanotube structure
makes it not suitable to obtain resonant antenna. Further theoretical and experimental
studies are still required to investigate the possibility of using parallel carbon nanotubes in
RF circuits and antennas. In addition electro-mechanical properties of carbon nanotubes can
be also be useful in RF applications like filtering and switching. On the other hand, the high
attenuation at lower frequencies below 100 GHz makes carbon nanotubes good candidate
for absorbing and shielding applications. Absorbing properties of carbon nanotubes can be
quite useful in medical applications such as microwave imaging and selective microwave
heating for cancer treatment. Thin film of carbon nanotubes above a polyethylene substrate
is also found to good candidate for transparent shielding surface up to 20dB. Higher
shielding effectiveness combined with high transparence is still under investigation.

10. Acknowledgement
The authors would like to acknowledge the assistance and the financial support provided by
the Research Center at the College of Engineering at King Saud University for this chapter.

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                                      Carbon Nanotubes Applications on Electron Devices
                                      Edited by Prof. Jose Mauricio Marulanda




                                      ISBN 978-953-307-496-2
                                      Hard cover, 556 pages
                                      Publisher InTech
                                      Published online 01, August, 2011
                                      Published in print edition August, 2011


Carbon nanotubes (CNTs), discovered in 1991, have been a subject of intensive research for a wide range of
applications. In the past decades, although carbon nanotubes have undergone massive research, considering
the success of silicon, it has, nonetheless, been difficult to appreciate the potential influence of carbon
nanotubes in current technology. The main objective of this book is therefore to give a wide variety of possible
applications of carbon nanotubes in many industries related to electron device technology. This should allow
the user to better appreciate the potential of these innovating nanometer sized materials. Readers of this book
should have a good background on electron devices and semiconductor device physics as this book presents
excellent results on possible device applications of carbon nanotubes. This book begins with an analysis on
fabrication techniques, followed by a study on current models, and it presents a significant amount of work on
different devices and applications available to current technology.



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Ahmed M. Attiya and Majeed A. Alkanhal (2011). Carbon Nanotubes in Passive RF Applications, Carbon
Nanotubes Applications on Electron Devices, Prof. Jose Mauricio Marulanda (Ed.), ISBN: 978-953-307-496-2,
InTech, Available from: http://www.intechopen.com/books/carbon-nanotubes-applications-on-electron-
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